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Parameter Estimation for the Square-root Diffusions : Ergodic and Nonergodic Cases
Mohamed Ben Alaya, Ahmed Kebaier
To cite this version:
Mohamed Ben Alaya, Ahmed Kebaier. Parameter Estimation for the Square-root Diffusions : Ergodic and Nonergodic Cases. 2010. �hal-00579644�
Parameter Estimation for the Square-root Diffusions : Ergodic and Nonergodic Cases
By Mohamed Ben Alaya and Ahmed Kebaier
LAGA, CNRS (UMR 7539), Institut Galil´ ee, Universit´ e Paris 13, 99, av. J.B. Cl´ ement 93430 Villetaneuse, France
[email protected] [email protected] March 24, 2011
Abstract
This paper deals with the problem of parameter estimation in the Cox-Ingersoll-Ross (CIR) model (Xt)t≥0. This model is frequently used in finance for example as a model for computing the zero-coupon bound price or as a dynamic of the volatility in the Heston model. When the diffusion parameter is known, the maximum likelihood estimator (MLE) of the drift parameters involves the quantities : ∫t
0Xsdsand∫t
0 ds
Xs. At first, we study the asymptotic behavior of these processes. This allows us to obtain various and original limit theorems on our estimators, with different rates and different types of limit distributions.
Our results are obtained for both cases : ergodic and nonergodic diffusion. Numerical simulations were processed using an exact simulation algorithm. AMS 2000 Math- ematics Subject Classification. 44A10, 60F05, 62F12, 65C05. Key Words and Phrases. Cox-Ingersoll-Ross processes, nonergodic diffusion, Laplace transform, limit theorems, parameter inference, simulation efficiency : exact methods.
1 Introduction
Over the last few years, an interesting process emerged and became quite popular in finance, after Cox-Ingersoll-Ross (CIR) proposed it for modelling short-term interest rates [4]. It is also used for modelling stochastic volatility in the Heston model [10]. The CIR process (Xt)t≥0, also known as the square root diffusion, is solution to the stochastic differential equation (SDE)
dXt = (a−bXt)dt+√
2σ|Xt|dWt, (1)
where X0 = x > 0, a > 0, b ∈ R, σ > 0 and (Wt)t≥0 is a standard Brownian motion. Under the above assumption on the parameters that we will suppose valid through all the paper, this SDE has a unique strong solution (Xt)t≥0 (see Ikeda and Watanabe [11], p. 221) and from the comparison theorem for one-dimensional diffusion process (see Revuz and Yor [20], p. 394)
we deduce that Xt ≥ 0. In the particular case b = 0 and σ = 2, we recover the square of a a-dimensional Bessel process starting at x. Let us recall now some basic properties on the CIR model and let τ0 := inf{t ≥ 0|Xt = 0}, with the convention inf∅=∞. In the case a≥ σ, the process is strictly positive,τ0 is infinite almost surely, otherwise it is nonnegative, which means that it can reach the state 0. More precisely, fora < σ and b≥0,τ0 is finite almost surely and for a < σ and b < 0, we have Px(τ0 < ∞)∈]0,1[. Note that in the case 0 < a < σ, when the process reaches the boundary, the state 0 is instantaneously reflecting (see e.g. G¨oing-Jaeschke and Yor [8] or Lamberton and Lapeyre [16] for more details). From the ergodicity point of view, the CIR process is ergodic and its stationary distribution, say π, is a Gamma law with shape parameter a/σ and scale parameter σ/b, provided that b > 0. In this case, for all h ∈ L1(π),
1 t
∫t
0 h(Xs)ds converges almost surely to ∫
Rh(x)π(dx).
During the last decades, several authors studied the problem of estimating parameters in the drift coefficient when a diffusion process was observed continuously; this corresponds to observe the path of the diffusion over an interval [0, T],T >0. This theory has been established mainly by Lipster and Shiryayev [17] and Kutoyants [15]. This approach is rather theoretical, since the real data are discrete time observations. However, if the error due to discretization is negligible then the statistical results obtained for the continuous time model are valid for discrete time observations too. In the literature, most of the articles are concerned with ergodic diffusions and only few results can be found for the nonergodic case (see e.g [15] section 3.5 of chapter 3 and references there). In this last reference, many technics are proposed to construct estimators of the drift parameters. Furthermore, when the drift coefficient depends linearly on the parameters, one can hope to obtain a nice explicit formula of the maximum likelihood estimator (MLE). To our knowledge, one of the first papers having studied the MLE for the problem of estimating parameters in the CIR model is that of Fourni´e and Talay [7]. They have established its asymptotic normality in the case b >0 and a > σ.
The aim of this paper is to investigate the MLE of the drift parameters in the CIR model for a range of values (a, b, σ) covering ergodic and nonergodic situations. Roughly speaking, if we estimate one of the drift parameters and suppose known the other one, the MLE error has the form Mt/hMit, where (Mt)t≥0 is a Brownian martingale with quadratic variationhMit. If b > 0 and a > σ, the asymptotic normality of the estimators is obtained using the classical martingale central limit theorem as in Fourni´e and Talay [7]. Otherwise, this argument is no more valid, even in the special ergodic case b > 0 and a ≤ σ, since ∫
R(1/x)π(dx) = ∞. To overcome this difficulty, we study the asymptotic behavior of the couple (Mt,hMit).
In the second section, as in our framework hMit is either ∫t
0 Xsds or ∫t 0
ds
Xs, we proceed by computing their Laplace transform. The first one is well known (see e.g. Lamberton and Lapeyre [16], p. 127). However for the second one, which is more subtle, we apply recent results of Craddock and Lennox [5], who employ Lie symmetry methods to evaluate certain expectations for a large class of Itˆo diffusions. This allows us to obtain a precise description of the asymptotic of the ∫t
0 Xsds (see Proposition 1 and 3) and ∫t
0 ds
Xs (see Proposition 2 and 4).
Then in the third section, we take advantage of this study to prove new original results on the asymptotic of the MLE that are not necessarily normal. The asymptotic theorem concerning the MLE, ˆbT (resp. ˆaT), of b (resp. a) is obtained with different rates of convergence that are unusual in most cases : √
T for b > 0, T for b = 0 and e−bT /2 for b < 0 (resp. √
T for b > 0 and a > σ, T for b >0 anda=σ,√
logT for b= 0 and a > σ and logT for b= 0 and a =σ).
In those different cases, the corresponding limit distributions are given by Theorem 1 for the MLE of b and Theorem 2 for the MLE of a.
Finally in the last section, we illustrate our asymptotic results using an exact simulation method. Indeed, to simulate the couple (XT,∫T
0 Xsds) we use and perfect the method of Broadie and Kaya [3] based on an explicit evaluation of the conditional Laplace transform of
∫T
0 Xsds given XT. Concerning (XT,∫T 0
ds
Xs), we first establish a new explicit formula of the conditional Laplace transform of ∫T
0 ds
Xs given XT (see Theorem 3 and 4) and then we deduce an exact simulation method of the couple in the same manner as in [3].
2 The Asymptotic Behavior of ∫
t0
X
sds and ∫
t 0ds Xs
Let us recall that (Xt)t≥0 denotes a CIR process solution to (1). It is relevant to consider separately the cases b= 0 andb 6= 0, since the process (Xt)t≥0 behaves differently.
2.1 Case b = 0
In this section, we consider the Cox-Ingersoll-Ross CIR process (Xt)t≥0 with b = 0. In this particular case, (Xt)t≥0 satisfies the SDE
dXt=adt+√
2σXtdWt. (2)
Note that for σ = 2, we recover the square of a a-dimensional Bessel process starting at x and denoted by BESQax. This process has been attracting considerably the attention of several studies (see Revuz and Yor [20]). The asymptotic behavior of (Xt,∫t
0 Xsds) and ∫t 0
ds Xs are established by our next two propositions.
Proposition 1 Let (Xt)t≥0 be a CIR process solution to (2), we have (Xt
t , 1 t2
∫ t
0
Xsds)−→law (R1, I1) as t tends to infinity.
where (Rt)t≥0 is the CIR process starting from 0, solution to (2) and It=∫t
0 Rsds.
Proposition 2 Under the above notations, we have 1. Px
(∫ t
0
ds Xs
<∞ )
= 1 if and only if a ≥σ.
2. If a > σ then 1 logt
∫ t 0
ds Xs
−→P 1
a−σ as t tends to infinity.
3. If a = σ then 1 (logt)2
∫ t
0
ds Xs
−→law τ1 as t tends to infinity, where τ1 is the hitting time associated with Brownian motion τ1 := inf{t >0 : Wt= √1
2σ}.
In order to prove these propositions, we choose to compute the Laplace transform of the couple (Xt,∫t
0 Xsds) and ∫t 0
ds
Xs. Here are the obtained results.
Lemma 1 We have
• For λ ≥0 and µ≥0, Ex
(
e−λXt−µR0tXsds )
=e−aφλ,µ(t)e−xψλ,µ(t), where functions φλ,µ and ψλ,µ are given by
φλ,µ(t) = 1 σlog
(2σλ
ρ sinh(ρt/2) + cosh(ρt/2) )
, ψλ,µ(t) = λcosh(ρt/2) + 2µρ sinh(ρt/2)
2σλ
ρ sinh(ρt/2) + cosh(ρt/2) and ρ= 2√σµ.
• For µ > 0, Ex
(
e−µR0tXsds )
= Γ(k+ν2 +12) Γ(ν+ 1)
(x σt
)ν
2+12−k
exp (−x
σt )
1F1 (
k+ ν 2 + 1
2, ν+ 1, x σt
) , (3) where k = a
2σ, ν = 1 σ
√(a−σ)2+ 4µσ and 1F1 is the confluent hypergeometric function defined by 1F1(u, v, z) =∑∞
n=0 un
vn
zn
n!, with u0 =v0 = 1, and for n ≥1, un =∏n−1
k=0(u+k) and vn =∏n−1
k=0(v+k).
Proof : Takingb= 0 in Proposition 2.5 of chapter 6 in [16], we deduce the first assertion. For the second one, we apply Theorem 5.10 in [5] to our process. We have just to be careful with the misprint in formula (5.24) of [5]. More precisely, we have to replace √
Axy, in the numerator of the first term in the right hand side of this formula, by√
Ax/y. Hence, fora >0 and σ >0, we obtain the so called fundamental solution of the PDEut=σxuxx+aux−(µx+λx)u, λ >
0, µ >0 :
p(t, x, y) =
√σλ σsinh(√
σλt) (y
x )k−1/2
exp (
−
√σλ(x+y) σtanh(√
σλt) )
Iν
( 2√ σλ√
xy σsinh(√
σλt) )
, (4) whereIν is the modified Bessel function of the first kind. This yields the Laplace transform of the couple (∫t
0 Xsds,∫t
0 ds
Xs), since Ex
(
e−λR0tXsds−µR0tXsds )
= ∫∞
0 p(t, x, y)dy. Evaluation of this integral is routine, see formula 2 of section 6.643 in [9]. Therefore, we get
Ex
( e−λ
Rt
0Xsds−µRt 0
ds Xs
)
= Γ(k+ ν2 +12) Γ(ν+ 1)
(√
σλxcoth(√ σλt) σ
)−k
exp (
−
√σλx
σ coth(√ σλt)
)
×exp
( √
σλx 2σsinh(√
σλt) cosh(√ σλt)
) M−k,ν
2
( √
σλx σsinh(√
σλt) cosh(√ σλt)
) , (5)
whereMs,r(z) is the Whittaker function of the first kind given by Ms,r(z) = zr+12e−z2 1F1(r−s+ 1
2,2r+ 1, z). (6)
See [9] for more details about those special functions. By inserting relation (6) in (5) we obtain
Ex
(
e−λR0tXsds−µR0tXsds )
= Γ(k+ ν2 +12) Γ(ν+ 1)
(
cosh(√ σλt)
)−ν2−12−k( √ σλx σsinh(√
σλt)
)ν2+12−k
×exp (
−
√σλx
σ coth(√ σλt)
)
1F1 (
k+ ν 2 + 1
2, ν+ 1,
√σλx σsinh(√
σλt) cosh(√ σλt)
) .
We complete the proof by lettingλ tend to 0.
Proof of Proposition 1 : Under the notations of the above Lemma, it is easy to check that
φλ
t,µ
t2(t) = −1 σlog
( √
σµ σλsinh(√
σµ) +√
σµcosh(√ σµ)
)
, lim
t→∞ψλ
t,µ
t2(t) = 0 and
tlim→∞Ex
(
e−λXtt −tµ2R0tXsds )
=
( √
σµ σλsinh(√
σµ) +√
σµcosh(√ σµ)
)a
σ
.
Noting that the first assertion of Lemma 1 remains valid with x= 0 (see [16]), we deduce that the obtained limit is simply the Laplace transform of the CIR process starting from 0, solution
to (2) witht = 1. This completes the proof.
Remark It is worth to note that Proposition 1 can be obtained by a scaling argument, but in order to standardize the technics used in this section we dropped this idea.
Proof of Proposition 2 Fora ≥σ, by Lemma 1, we have Px
(∫ t 0
ds Xs <∞
)
= lim
µ→∞Ex
(
e−µR0tXsds )
= 1.
In the case a < σ, we have Px
(∫ t 0
ds Xs <∞
)
= 1
Γ(2− aσ) (x
σt )1−aσ
exp (−x
σt )
1F1
(
1,2− a σ, x
σt )
.
Thanks to the following formula (see section 9.211 in [9])
1F1(r, s, z) = Γ(s)
Γ(r)Γ(s−r)z1−s
∫ z
0
euur−1(z−u)s−r−1du,
for Re(s)>Re(r)>0, we obtain Px
(∫ t 0
ds Xs <∞
)
= e−σtx Γ(1− σa)
∫ x
σt
0
eu (x
σt −u )−a
σ du.
After the change of variable, v = σtx −u, the last relation becomes Px
(∫ t
0
ds Xs <∞
)
= 1
Γ(1−σa)
∫ x
σt
0
e−vv−aσdv <1.
For the second and the third assertions, we consider a positive function γ(t) increasing to +∞ when t→+∞. Using standard evaluations, it is easy to prove that
t→lim+∞Ex
( e−
µ γ(t)2
Rt 0
ds Xs
)
= lim
t→+∞exp (
− 1 2σ
(
σ−a+
√
(a−σ)2+ 4µσ γ(t)2
) log (t)
) .
• Ifa > σ, let ε denotes a function such that limx→0ε(x) = 0, we have
t→lim+∞Ex
( e−
µ γ(t)2
Rt 0
ds Xs
)
= lim
t→+∞exp (
−
( µ
γ(t)2(a−σ)+ 1 γ(t)2ε
( 1 γ(t)2
) log (t)
))
= exp (
− µ a−σ
)
, by taking γ(t)2 = log(t).
• Ifa=σ
t→+∞lim Ex
( e−
µ γ(t)2
Rt 0
ds Xs
)
= lim
t→+∞exp (
− 1
√σ
õ
γ(t)log (t) )
= exp (
−
õ
√σ )
, by taking γ(t) = log(t).
This completes the proof.
We now turn to the case b6= 0.
2.2 Case b 6 = 0
Let us resume the general model of the CIR given by relation (1) withb 6= 0, namely dXt = (a−bXt)dt+√
2σXtdWt, (7)
where X0 =x > 0, a > 0, b ∈R∗, σ > 0. Note that this process may be represented in terms of a square Bessel process through the relation Xt = e−btY (σ
2b(ebt−1))
, where Y denotes a BESQ
2a
xσ . This relation results from simple properties of square Bessel processes (see e.g.
G¨oing-Jaeschke and Yor [8] and Revuz and Yor [20]). We can now formulate the main results of this subsection.
Proposition 3 Let (Xt)t≥0 be a CIR process solution to (7), we have 1. If b >0 then 1
t
∫ t
0
Xsds−→P a
b as t tends to infinity.
2. Ifb <0then (
ebtXt, ebt
∫ t 0
Xsds )
−→law (Rt0, t0Rt0), asttends to infinity, wheret0 =−1/b and (Rt)t≥0 is the CIR process, starting from x, solution to (2).
Proposition 4 Under the above notations, we have 1. Px
(∫ t 0
ds Xs <∞
)
= 1 if and only if a ≥σ.
2. If b >0 and a > σ then 1 t
∫ t
0
ds Xs
−→P b
a−σ as t tends to infinity.
3. If b > 0 and a = σ then 1 t2
∫ t
0
ds Xs
−→law τ2 as t tends to infinity, where τ2 is the hitting time associated with Brownian motion τ2 := inf{t >0 : Wt= √b
2σ}. 4. Ifb <0anda≥σ then
∫ t 0
ds Xs
−→law It0 :=
∫ t0
0
Rsds asttends to infinity, wheret0 =−1/b and (Rt)t≥0 is the CIR process, starting from x, solution to (2).
Remark : When a > σ and b >0 the CIR process is ergodic and the stationary distribution is a Gamma law with shapea/σ and scale σ/b. Let ξlaw= Γ(a/σ, σ/b), according to the ergodic theorem, 1t∫t
0 Xsds P−−→p.s. E(ξ) = ab and 1t∫t 0
ds Xs
P−p.s.
−→ E(1ξ) = a−bσ as t tends to infinity. In this case we recover the first assertion of Proposition 3 and the second assertion of Proposition 4.
In order to prove these propositions we need the following result.
Lemma 2 We have,
• for λ≥0 and µ≥0, the Laplace transform of (Xt,∫t
0 Xsds) is given by Ex
(
e−λXt−µR0tXsds )
=e−aφ˜λ,µ(t)e−xψ˜λ,µ(t), (8) where
φ˜λ,µ(t) = −1 σlog
( 2ρet(b−ρ)/2
2σλ(1−e−ρt) + (ρ−b)e−ρt+ (ρ+b) )
and
ψ˜λ,µ(t) = λ((ρ+b)e−ρt+ (ρ−b)) + 2µ(1−e−ρt) 2σλ(1−e−ρt) + (ρ−b)e−ρt+ (ρ+b) , with ρ=√
b2+ 4σµ.
• For µ > 0, the Laplace transform of ∫t 0
ds
Xs is given by Ex
(
e−µR0tXsds )
= Γ(k+ ν2 + 12) Γ(ν+ 1)
1
xkαkβν2+12
×exp ( b
2σ [
at− 2x ebt−1
])
1F1 (
k+ν 2 +1
2, ν+ 1, β )
,
(9)
where k = a
2σ, α= bebt
σ(ebt−1), β = bx
σ(ebt−1) and ν= 1 σ
√(a−σ)2+ 4µσ.
Remark : Note that the limit of the above Laplace transform, when b goes to 0, in formula (9) allows us to recover relation (3).
Proof The first assertion is given by Proposition 2.5 of chapter 6 in [16]. For the second one, we first apply Theorem 5.7 of [5], for b ∈ R. We obtain the fundamental solution of the PDE ut=σxuxx+ (a−bx)ux− µxu, µ >0 :
p(t, x, y) = |b| 2σsinh(|b|t/2)
(y x
)a/(2σ)−1/2
×exp ( b
2σ[at+ (x−y)]− |b|(x+y) 2σtanh(|b|t/2)
) Iν
( |b|√xy σsinh(|b|t/2)
) .
(10)
By the parity of hyperbolic functions, we omit the |.| in the above formula. This yields the Laplace transform of ∫t
0 ds
Xs, since Ex
(
e−µR0tXsds )
= ∫∞
0 p(t, x, y)dy. In the same manner as in the proof of Lemma 1, formula 2 of section 6.643 in [9] gives us
Ex
(
e−µR0tXsds )
= Γ(k+ ν2 +12) Γ(ν+ 1)
eβ2 xkαk exp
( b 2σ
[
at− 2x ebt−1
]) M−k,ν
2 (β). (11)
Finally, by inserting relation (6) in (11) we obtain the announced result.
Remark : In the above proof, relation (10) extends Corollary 5.8 of [5], established in the case b >0, to the case b∈R. It is worth to note that formula (5.20) in this Corollary remains valid for this extension, thanks to the parity of hyperbolic functions.
In the following proofsε will denotes a function satisfying limx→0ε(x) = 0, that can change from an evaluation to an other.
Proof of Proposition 3 Let γ(t) be a (non-random) positive function increasing to +∞ when t → +∞, we take λ = 0 and replace µ by µ/γ(t)2 in relation (8). In the caseb > 0, an easy computation shows that
t→lim+∞Ex
( e−
µ γ(t)2
Rt 0Xsds)
= lim
t→+∞exp (
−ab 2σt
(√
1 + 4µσ b2γ(t)2 −1
))
= exp (
−ab 2σ
t γ(t)2
(2µσ
b2 +ε( 1 γ(t)2)
)) .
The first assertion follows by choosing γ(t)2 =t.
Next, we study the case b <0. According to the first assertion of Lemma 2, we have Ex
(
e−λebtXt−µebtR0tXsds )
=e−aφ˜λebt,µebt(t)e−xψ˜λebt,µebt(t) with ρ = √
b2+ 4σµebt. As the Taylor’s expansion of ρ+b is equal to −2σµb ebt +ebtε(ebt), we deduce that limt→+∞(ρ+b)t = 0 and limt→+∞(ρ+b)eρt = limt→+∞(ρ+b)e−bt =−2σµb . Hence, it’s easy to check that
t→lim+∞φ˜λebt,µebt(t) =−1 σlog
( −b σλ−b− σbµ
)
and lim
t→+∞ψ˜λebt,µebt(t) = −bλ+µ σλ−b− σbµ.
Therefore
t→lim+∞Ex
(
e−λebtXt−µebtR0tXsds )
=
( −b σλ−b− σbµ
)a/σ
exp (
−x −bλ+µ σλ−b−σbµ
)
In the other hand, using the first assertion of Lemma 3, we identify the Laplace transform of the announced couple limit (Rt0, t0Rt0), which completes the proof.
Remark : For b < 0, we can also give this representation ebt
∫ t 0
Xsds −→law L := L1 + L2 as t tends to infinity where L1 and L2 are two independent random variables, L1 has a gamma distribution and L2 has a compound Poisson distribution. More precisely L1 law= Γ(a/σ, σ/b2) and L2 law= ∑N
k=1Xk where N law= P(xb/σ), for all k ≥ 1, Xk law= E(σ/b2) and (N, X1,· · · , Xn,· · ·) are mutually independent.
Proof of Proposition 4 Note that, the Laplace transform of ∫t
0 ds
Xs converges toPx(∫t
0 ds Xs <
∞) as µ→0. If a ≥σ, using standard evaluations, it is easy to prove that Px
(∫ t 0
ds Xs <∞
)
= ( β
xα )a
2σ
exp ( b
2σ [
at− 2x ebt−1
] +β
)
= 1.
In the other case, a < σ, we have Px
(∫ t
0
ds Xs
<∞ )
= 1
Γ(2− aσ) β1−2σa (xα)2σa exp
( b 2σ
[
at− 2x ebt−1
])
1F1 (
1,2− a σ, β
)
= β1−aσ
Γ(2− aσ)e−β1F1 (
1,2− a σ, β
) .
Thanks to the following formula (see section 9.211 in [9])
1F1(r, s, z) = Γ(s)
Γ(r)Γ(s−r)z1−s
∫ z
0
euur−1(z−u)s−r−1du,
for Re(s)>Re(r)>0, we obtain Px
(∫ t
0
ds Xs <∞
)
= e−β Γ(1−σa)
∫ β
0
eu(β−u)−σa du.
After the change of variable, v =β−u, the last relation becomes Px
(∫ t 0
ds Xs <∞
)
= 1
Γ(1− aσ)
∫ β 0
e−vv−σadv <1.
Now our task is to study the asymptotic behavior in distribution of the quantity ∫t
0 ds Xs
when b >0 and a≥ σ. Let γ(t) be a (non-random) positive function increasing to +∞ when t→+∞. If we replaceµbyµ/γ(t)2 in relation (9), since log(β) = log(bx/σ)−bt−log(1−e−bt),
t→lim+∞Ex
( e−
µ γ(t)2
Rt 0
ds Xs
)
= lim
t→+∞
1 xkαk exp
( ab 2σt−
[ 1 2σ
√
(a−σ)2+ 4µσ γ(t)2 + 1
2 ]
log(β) )
= lim
t→+∞exp (
ab 2σt−
[ 1 2σ
√
(a−σ)2+ 4µσ γ(t)2 + 1
2 ]
bt )
.
• Fora > σ, we have
t→lim+∞Ex
( e−
µ γ(t)2
Rt 0
ds Xs
)
= lim
t→+∞exp (
ab 2σt−
[ a−σ
2σ
√
1 + 4µσ
γ(t)2(a−σ)2 +1 2 ]
bt )
= lim
t→+∞exp
( µbt
γ(t)2(a−σ)+ t
γ(t)2ε( 1 γ(t)2)
) .
Taking γ(t)2 = t, we deduce that 1t ∫t
0 ds
Xs converges in distribution and of course in probability to the constant b/(a−σ).
• Fora=σ, we have
t→lim+∞Ex
( e−
µ γ(t)2
Rt 0
ds Xs
)
= lim
t→+∞exp (
− bt
√σγ(t)
õ )
.
Taking now γ(t) =t, we deduce that t12
∫t 0
ds
Xs converges in distribution to τ2.
It remains now to prove the last assertion. Forb <0, whent goes to infinity in relation (9), since α converges to 0 and β to−bx/σ, we have
t→+∞lim Ex
( e−µ
Rt 0
ds Xs
)
= Γ(k+ ν2 + 12) Γ(ν+ 1)
(−bx σ
)ν2+12
1 xk 1F1
( k+ ν
2 + 1
2, ν+ 1,−bx σ
)
× lim
t→+∞exp ( b
2σ [at+ 2x]−klogα )
= Γ(k+ ν2 + 12) Γ(ν+ 1)
(−bx σ
)ν2+12−k
exp (bx
σ )
1F1
( k+ν
2 +1
2, ν+ 1,−bx σ
) .
Finally, we conclude by identifying the limit distribution with relation (3) in Lemma 1.
3 Statistical Inference of the CIR model
Let us first recall some basic notions on the construction of the maximum likelihood estimator (MLE). Suppose that the one dimensional diffusion process (Xt)t≥0 satisfies
dXt =b(θ, Xt)dt+σ(Xt)dWt, X0 =x0,
where the parameter θ ∈ Θ⊂ Rp, p ≥1, is to be estimated. The coefficients b and σ are two functions satisfying conditions that guarantee the existence and uniqueness of the SDE for each θ∈Θ. We denote by Pθ the probability measure induced by the solution of the equation on the canonical space C(R+,R) with the natural filtration Ft :=σ(Ws, s ≤t), and let Pθ,t :=Pθ|Ft
be the restriction of Pθ to Ft. If the integrals in the next formula below make sense then the measures Pθ,t and Pθ0,t, for any θ, θ0 ∈Θ, t >0, are equivalent (see Jacod [12] and Lipster and Shirayev [17]) and we are able to introduce the so called likelihood ratio
Lθ,θt 0 := dPθ,t
dPθ0,t
= exp {∫ t
0
b(θ, Xs)−b(θ0, Xs)
σ2(Xs) dXs− 1 2
∫ t
0
b2(θ, Xs)−b2(θ0, Xs) σ2(Xs) ds
}
. (12) The process
( Lθ,θt 0
)
t≥0 is an Ft−martingale.
In the present section, we observe the process XT = (Xt)0≤t≤T as a parametric model solution to equation (1), namely
dXt = (a−bXt)dt+√
2σXtdWt,
whereX0 =x >0,a >0,b∈R,σ > 0. The unknown parameter, sayθ, is involved only in the drift part of the diffusion and we consider the two cases θ =b or θ = a. This study includes both ergodic (b >0 and a≥σ) and nonergodic cases.
3.1 Parameter estimation θ = b
The appropriate likelihood ratio (12), evaluated at time T with θ0 = 0, is well defined and is given by
LT(b) :=Lθ,θT 0 = exp { b
2σ(x−XT) + 1 4σ
∫ T
0
(2ab−b2Xs)ds }
.
The MLE ˆbT of b maximizes LT(b), then
ˆbT = aT +x−XT
∫T
0 Xsds . (13)
Hence, the error is given by
ˆbT −b =−√ 2σ
∫T
0
√XsdWs
∫T
0 Xsds .
As mentioned in the introduction, the above error is obviously of the form Mt/hMit, where (Mt)t≥0 is a Brownian martingale with quadratic variation hMit. If this quadratic variation,
correctly normalized, converges in probability then the classical martingale central limit theo- rem can be applied. Otherwise, the study of the couple (XT,∫T
0 Xs) will be helpful to investigate the limit law of the error. The asymptotic behavior of ˆbT −b can be summarized as follows.
Theorem 1 The MLE of b satisfies 1. Case b >0 : Lb
{√
T(ˆbT −b) }
=⇒ N(0,2σb a).
2. Caseb = 0 : Lb
{
T(ˆbT −b) }
=⇒ a−R1
I1 , where (Rt) is the CIR process, starting from0, solution to (2) and It =∫t
0 Rsds.
3. Case b < 0 : Lb
{
e−bT /2(ˆbT −b) }
=⇒ G
R, where (G, R) is a couple of random variable characterized with its joint moment generating-Laplace transform. For λ∈R and µ≥0,
E(
eλG−µR)
=
( b µσ/b+b
)σa exp
(
xσλ2/b+µ µσ/b+b
) .
ThereforeG and R are correlated, G is normal andR has the same distribution as t0Rt0, t0 =−1/b, where (Rt)t≥0 is the CIR process, starting from x, solution to (2).
Proof : In the case b > 0, by Proposition 3 and the central limit theorem given by Y.A.
Kutoyants (see Theorem 1.19 in [15]) we have Pb− lim
T→∞
1 T
∫ T
0
Xsds= a b, Lb
{ 1
√T
∫ T
0
√XsdWs }
=⇒ N(0,a b).
Therefore we obtain the first assertion. In the second caseb = 0, by Proposition 1 we have L0
{XT T , 1
T2
∫ T
0
Xsds }
=⇒(R1, I1).
Hence
L0
{
T(ˆbT −b) }
=L0
{a+Tx − XTT
1 T2
∫T 0 Xsds
}
=⇒ a−R1
I1 .
For the last case b <0, by Proposition 3 we have Lb
{
ebTXT, ebT
∫ T
0
Xsds }
=⇒(
R˜t0, t0R˜t0 )
,
where t0 = −1/b and ( ˜Rt)t≥0 is the CIR process, starting from x, solution to (2). It follows that
Lb
{ˆbT −b }
=Lb
{ebT(aT +x−XT −b∫T
0 Xsds) ebT ∫T
0 Xsds
}
=⇒0
